Color management: Monitor calibration issues + Compositor issues (+ ideas for improvement)
Main issues currently bothering me:

There are quite a few issues, and thankfully more folks are learning them.

The final render is displayed in my monitor's color profile, which is nice. However, after saving the file is in this profile as well, i.e. I have to process it in another step...

This is because Blender has a horrific file encoding chain. The default is to bake in the Display / View / Look transform, which means that if you have any colourimetric twists in your displays, that will end up encoded into the code values.
Hack Solution:

For canonical sRGB encoding to nonlinear display referred outputs, prior to saving to a display referred output such as JPEG or TIFF, change the Display / View / Look combination to the same combination with sRGB chosen as display.
For EXR encoding, the native buffer will be encoded, so the results skip all management altogether.

Proper Solution:

Pressure the development to implement an encoding panel. This would be something akin to a selection box with a stack of elements similar to a texture, that would have sane defaults and allow the audience to encode a specific chain of transforms into their output imagery.


Compositor: If I load a file, which should be on top of the render (model on a green-screen, logo, anything, ...) it is processed by the same color transformation as the render: Filmic + sRGB -&gt; monitor's profile. The latter part is ok I guess, but Filmic should not be applied here. How to turn it off for images imported in compositor?

When you load an image, you must describe the encoding to Blender. Sadly, this is hidden away behind a bunch of obfuscating menu elements. One method is to open up the Image Viewer, open the sidebar properties panel on the right, and select the encoding that describes the encoding of the footage or shot.
This will cascade into another problem because, again, generic imagery does not work in a radiometric-like compositing environment. There's nothing short of hackery and quackery that can be done to try and get a generic sRGB image into the radiometric domain, as he data is not there.
Hack Solution:

Set the sRGB image to be Filmic encoded. This will artificially &quot;stretch&quot; the range to become a no-operation after the output chain.

More Proper Solution:

Use properly encoded imagery! Camera raws, log encodings with documented transforms, etc. can all be coaxed to do things more correctly. For all work generated from within Blender, never ever use any other file encoding other than EXR. Start with EXR. Stay there. Never leave. Only after completely finished, deliver to the delivery encodings.

It's a deeper subject than this, but hopefully there's enough here to get you moving in the right direction.